The hypergroupoid of boundary conditions for local quantum observables

Marcel Bischoff, Karl-Henning Rehren
December 09, 2016
We review the definition of hypergroups by Sunder, and we associate a hypergroup to a type III subfactor $N\subset M$ of finite index, whose canonical endomorphism $\gamma\in\mathrm{End}(M)$ is multiplicity-free. It is realized by positive maps of $M$ that have $N$ as fixed points. If the depth is $>2$, this hypergroup is different from the hypergroup associated with the fusion algebra of $M$-$M$ bimodules that was Sunder's original motivation to introduce hypergroups. We explain how the present hypergroup, associated with a suitable subfactor, controls the composition of transparent boundary conditions between two isomorphic quantum field theories, and that this generalizes to a hypergroupoid of boundary conditions between different quantum field theories sharing a common subtheory.
open access link
@article{Bischoff:2016rpu, author = "Bischoff, Marcel and Rehren, Karl-Henning", title = "{The hypergroupoid of boundary conditions for local quantum observables}", journal = "Adv. Stud. Pure Math.", volume = "80", year = "2019", pages = "32-42", eprint = "1612.02972", archivePrefix = "arXiv", primaryClass = "math-ph", SLACcitation = "%%CITATION = ARXIV:1612.02972;%%" }

algebraic quantum field theory, hypergroups, defect, boundary conditions